Open Access
October 2019 Linear hypothesis testing for high dimensional generalized linear models
Chengchun Shi, Rui Song, Zhao Chen, Runze Li
Ann. Statist. 47(5): 2671-2703 (October 2019). DOI: 10.1214/18-AOS1761


This paper is concerned with testing linear hypotheses in high dimensional generalized linear models. To deal with linear hypotheses, we first propose the constrained partial regularization method and study its statistical properties. We further introduce an algorithm for solving regularization problems with folded-concave penalty functions and linear constraints. To test linear hypotheses, we propose a partial penalized likelihood ratio test, a partial penalized score test and a partial penalized Wald test. We show that the limiting null distributions of these three test statistics are $\chi^{2}$ distribution with the same degrees of freedom, and under local alternatives, they asymptotically follow noncentral $\chi^{2}$ distributions with the same degrees of freedom and noncentral parameter, provided the number of parameters involved in the test hypothesis grows to $\infty$ at a certain rate. Simulation studies are conducted to examine the finite sample performance of the proposed tests. Empirical analysis of a real data example is used to illustrate the proposed testing procedures.


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Chengchun Shi. Rui Song. Zhao Chen. Runze Li. "Linear hypothesis testing for high dimensional generalized linear models." Ann. Statist. 47 (5) 2671 - 2703, October 2019.


Received: 1 June 2017; Revised: 1 July 2018; Published: October 2019
First available in Project Euclid: 3 August 2019

zbMATH: 07114925
MathSciNet: MR3988769
Digital Object Identifier: 10.1214/18-AOS1761

Primary: 62F03
Secondary: 62J12

Keywords: High dimensional testing , likelihood ratio statistics , linear hypothesis , score test , Wald test

Rights: Copyright © 2019 Institute of Mathematical Statistics

Vol.47 • No. 5 • October 2019
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